A particle of mass m moving with a speed v hits elastically another stationary particle of mass 2m on a smooth horizontal circular tube of radius r. The time in which the next collision will take place is equal to


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Question Number 24945 by Tinkutara last updated on 29/Nov/17

$A particle of mass m moving with a$ $speed v hits elastically another$ $stationary particle of mass 2 m on a$ $smooth horizontal circular tube of$ $radius r . The time in which the next$ $collision will take place is equal to$
Commented by Tinkutara last updated on 30/Nov/17

$Thank you Sir!$
Commented by mrW1 last updated on 29/Nov/17

$s i n c e t h e h i t i s w i t h o u t e n e r g y l o s t,$ $t h e d e p a r t i n g v e l o c i t y a f t e r t h e h i t$ $i s e q u a l t o t h e a p p r o c h i n g v e l o c i t y$ $b e f o r e t h e h i t, w h i c h i s v .$ $i . e . a f t e r t h e h i t b o t h p a r t i c l e s$ $d e p a r t f r o m e a c h o t h e r w i t h v i n$ $t h e t u b e . t h e t i m e a f t e r w h i c h t h e y h i t$ $a g a i n i s h e n c e \frac{2 π r}{v} .$
Commented by mrW1 last updated on 29/Nov/17

$I t h i n k i t {d o e s n}^{'} t m a t t e r w h i c h m a s s$ $t h e y h a v e .$
Commented by mrW1 last updated on 30/Nov/17

$t h i s i s t h e p r o o f :$ $l e t m_{2} = λ m_{1}$ $b e f o r e t h e h i t :$ $p a r t i c l e 1 : v$ $p a r t i c l e 2 : 0$ $a f t e r h i t :$ $p a r t i c l e 1 : u_{1}$ $p a r t i c l e 2 : u_{2}$ $m_{1} v = m_{1} u_{1} + m_{2} u_{2}$ $\Rightarrow v = u_{1} + λ u_{2} . . . (i)$ $\frac{1}{2} m_{1} v^{2} = \frac{1}{2} m_{1} u_{1}^{2} + \frac{1}{2} m_{2} u_{2}^{2}$ $\Rightarrow v^{2} = u_{1}^{2} + λ u_{2}^{2} . . . (i i)$ $(i) i n t o (i i)$ $\Rightarrow v^{2} = v^{2} - 2 λ {v u}_{2} + λ^{2} u_{2}^{2} + λ u_{2}^{2}$ $\Rightarrow λ u_{2} [(λ + 1) u_{2} - 2 v] = 0$ $\Rightarrow (λ + 1) u_{2} = 2 v$ $\Rightarrow u_{2} = \frac{2}{λ + 1} \times v$ $\Rightarrow u_{1} = v - λ \times \frac{2 v}{λ + 1} = \frac{1 - λ}{λ + 1} \times v$ $\Rightarrow Δ u = u_{2} - u_{1} = \frac{2 - 1 + λ}{λ + 1} \times v = v$ $i . e . a f t e r h i t t h e p a r t i c l e s d e p a r t f r o m$ $e a c h o t h e r w i t h t h e s a m e v e l o c i t y$ $a s t h e y a p p r o c h t o e a c h o t h e r b e f o r e$ $t h e h i t .$ $t h e t i m e t i l l n e x t c o l l i s i o n i s h e n c e$ $\frac{2 π r}{v}$
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